2024 AIME I Problems/Problem 6
Problem
Consider the paths of length that follow the lines from the lower left corner to the upper right corner on an
grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.
Solution
We divide the path into eight “” movements and eight “
” movements. Five sections of alternative
or
are necessary in order to make four “turns.” We use the first case and multiply by
.
For , we have seven ordered pairs of positive integers
such that
.
For , we subtract
from each section (as the minimum is
) and we use Stars and Bars to get
.
Thus our answer is .